Question

find the number of terms in the arithmetic series where \\(a_1 = 19\\), \\(a_n = 118\\), and \\(s_n = 822\\)

question 2
find the number of terms in the arithmetic series where \\(a_1 = -2\\), \\(a_n = -62\\), and \\(s_n = -224\\).

Explanation:

Identify the given values for Question 1

For the first question, we are given:

• First term \(a_1 = 19\)
• Last term \(a_n = 118\)
• Sum of the series \(S_n = 822\)

Apply the arithmetic series sum formula for Question 1

The sum of an arithmetic series is given by:

$$S_n = \frac{n}{2}(a_1 + a_n)$$

Substitute the given values into the formula:

$$822 = \frac{n}{2}(19 + 118)$$

Solve for n in Question 1

Simplify the equation to find the number of terms:

$$822 = \frac{n}{2}(137)$$

$$1644 = 137n$$

$$n = \frac{1644}{137}$$

$$n = 12$$

Identify the given values for Question 2

For the second question, we are given:

• First term \(a_1 = -2\)
• Last term \(a_n = -62\)
• Sum of the series \(S_n = -224\)

Solve for n in Question 2

Substitute the values into the sum formula:

$$-224 = \frac{n}{2}(-2 + (-62))$$

$$-224 = \frac{n}{2}(-64)$$

$$-224 = -32n$$

$$n = \frac{-224}{-32}$$

$$n = 7$$

Answer:

Question 1

Find the number of terms in the arithmetic series where \(a_1 = 19\), \(a_n = 118\), and \(S_n = 822\) <blank>12</blank>

Question 2

Find the number of terms in the arithmetic series where \(a_1 = -2\), \(a_n = -62\), and \(S_n = -224\). <blank>7</blank>